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On J-Colorability of Certain Derived Graph Classes

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On J-Colorability of Certain Derived Graph Classes On J-Colorability of Certain Derived Graph Classes Federico Fornasiero Department of mathemathic Universidade Federal de Pernambuco Recife, Pernambuco, Brasil [email protected] Sudev Naduvath Centre for Studies in Discrete Mathematics Vidya Academy of Science & Technology Thrissur, Kerala, India. [email protected] Abstract A vertex v of a given graph G is said to be in a rainbow neighbourhood of G, with respect to a proper coloring C of G, if the closed neighbourhood N[v] of the vertex v consists of at least one vertex from every colour class of G with respect to C. A maximal proper colouring of a graph G is a J-colouring of G if and only if every vertex of G belongs to a rainbow neighbourhood of G. In this paper, we study certain parameters related to J-colouring of certain Mycielski type graphs. Key Words: Mycielski graphs, graph coloring, rainbow neighbourhoods, J-coloring arXiv:1708.09798v2 [math.GM] 4 Sep 2017 of graphs. Mathematics Subject Classification 2010: 05C15, 05C38, 05C75. 1 Introduction For general notations and concepts in graphs and digraphs we refer to [1, 3, 13]. For further definitions in the theory of graph colouring, see [2, 4]. Unless specified otherwise, all graphs mentioned in this paper are simple, connected and undirected graphs. 1 2 On J-colorability of certain derived graph classes 1.1 Mycielskian of Graphs Let G be a triangle-free graph with the vertex set V (G) = fv1; : : : ; vng. The Myciel- ski graph or the Mycielskian of a graph G, denoted by µ(G), is the graph with ver- tex set V (µ(G)) = fv1; v2; : : : ; vn; u1; u2; : : : ; un; wg such that vivj 2 E(µ(G)) () vivj 2 E(G), viuj 2 E(µ(G)) () vivj 2 E(G) and uiw 2 E(µ(G)) for all i = 1; : : : ; n. w u1 u2 u3 u4 u5 u6 u7 v1 v2 v3 v4 v5 v6 v7 Figure 1: The Mycielski graph µ(P7) In the above mentioned conditions of Mycielski graphs, we call the two vertices vi; ui twin vertices and the vertex w is called the root vertex of the Mycielskian µ(G). By a Mycielski type graph, we mean a graph that can be constructed from the Mycielski graphs or the graphs generated from a given graphs using some or similar rules of constructing their Mycielski graphs. 1.2 J-Coloring of Graphs The notion of J-coloring of a graph, has been defined for the first time in [10] as follows: Definition 1.1. [10] A graph G is said to have a J-colouring C if it has the maximal number of colours such that if every vertex v of G belongs to a rainbow neighbour- hood of G. The number of colours in a J-coloring C of G is called the J-colouring number of G. Definition 1.2. [10] A graph G is said to have a J ∗-colouring C if it has the maximal number of colours such that if every internal vertex v of G belongs to a rainbow neighbourhood of G. The number of colours in a J ∗-coloring C of G is called the J ∗-colouring number of G. It can be noted that all graphs, in general, need not have a J-coloring. Hence, the studies on the graphs which admit J-coloring and their properties and structural characterisations attract much interests. Some studies in this direction can be seen in [6, 10]. The initial purpose of this paper is to study the J-colourability of the Myciel- skian and certain Mycielski type graphs of some fundamental graph classes. Federico Fornasiero and Sudev Naduvath 3 2 J-Colourability of Mycielski Graphs Note that the Mycielski graph µ(G) of a graph G has no pendant vertices and hence the J-colouring and the J ∗-colouring of the Mycielski graphs µ(G), if they exist, are the same. We first try to repeat the original demonstration of Mycielskian graph. To do that, we have to fix a J-colouring on the graph G. Theorem 2.1. Let G be a graph with J-colouring C = fc1; c2; c3 : : : ; ckg. Then, the graph µ(G) is not J-colourable (and so, it is not J ∗-colourable). Proof. Note that µ(K2) is isomorphic to C5 and hence it is not J −colourable (as it is proved in [10]). Hence, we can consider graphs with order greater than 2. Let us assume that we can have a new J-colouring C = fc1; c2 : : : ; cjg on µ(G). Without 0 loss of generality, we can assume now that the colour of w is c1. Every vertex ui have to be coloured with one of the other colour c2; : : : ; cj. Hence, there exists at least a vertex v1 with colour c1. Let be v2 the vertex connected to v1 such as the twin vertex u2 has the colour c2. Here, we have the following two possibilities: (i) : If the colour of v2 is different from the previous ones, let us say c3, we have that for the definition of rainbow neighbourhood even the twin vertex u2 has to be connected with a vertex with the same colour, and it has to be a vertex v3 because no one of the vertices uj are connected, and w has the colour c1. But if it is so, than for the construction of µ(G) even the vertex v2 has to be connected with v3, and so it is not a proper coloring because two vertices has the same colours (see Figure 2). w c1 u1 c2 c2 u2 v3 c3 v1 c1 c3 v2 Figure 2: Case (i) (ii) If the colour of v2 is c2, the twin vertices u1 of v1 has to have a different colour 0 (let us say c3), because it is linked to w that has the colour c1 and to v2 that has the colour c2. But, in this case the vertex v1 has to be connected with another vertex which has the colour c3. So we have to differentiate two different possibilities: (ii)(a) If the vertex v1 is connected to a vertex v3 who has the colour c3 for the construction of µ(G) even the twin vertices u1 is connected to v3 and so it 4 On J-colorability of certain derived graph classes is not a proper coloring because two connected vertices have the same colour (see Figure 3). w c1 c3 c2 u1 u2 c3 c1 c2 v3 v1 v2 Figure 3: Case (ii)(a) (ii)(b) If the vertex v1 is connected to a vertex u3 which has the colour c3, first we can note that the twin vertex v3 cannot have a new colour, because it would lead to a contradiction over u3 similar to the case (i). Note that v3 cannot have the colour c1 (because for construction it is connected with v1) nor the colour c3 (because always for construction it is connected to the twin vertex u1) so it has to be coloured with the colour c2. But if it is so, u3 needs to be connected with a vertex vi whose colour is c2, but it cannot be the twin vertex v3 for the construction, nor the vertex v2 because it will lead to have the triangle v1v2u3v1. If the graph G has only 3 nodes we reach a contradiction yet, if it is not let us call v4 the vertex with colour c2 connected to u3. For the construction of µ(G) it has to be connected to the vertex v3 and it finally leads to a contradiction because the two vertices would have the same colour (see Figure 4). w c1 c2 c3 c3 u2 u1 u3 c2 c1 c2 c2 v2 v1 v3 v4 Figure 4: Case (ii)(b) Hence, the Mycielskian graph of any graph G is not J-colourable, irrespective of whether the G is J-colourable or not. Federico Fornasiero and Sudev Naduvath 5 3 Some New Constructions Since Mycielskian of any graph does not admit a J-colouring, our immediate aim is to construct some simple connected graphs from certain given graphs such that new graphs also admit an extended J-coloring. In this section, we discuss the J-colorability of certain newly constructed Mycielski type graphs of a given graphs. The first one among such graphs is the crib graph, denoted by c(G), of a graph G, which is defined in [11] as follows: Definition 3.1. [11]The crib graph, denoted by c(G), of a graph G is the graph whose vertex set is V (µ(G)) = fv1; v2; : : : ; vn; u1; u2; : : : ; un; wg such that vivj 2 E(µ(G)) () vivj 2 E(G), viuj 2 E(µ(G)) () vivj 2 E(G) and viw; uiw 2 E(µ(G)) for all i = 1; : : : ; n. Figure 5 depicts the crib graph of P6. w u1 u2 u5 u6 u3 u4 v1 v2 v3 v4 v5 v6 Figure 5: Crib graph of P6 The following theorem discusses the admissibility of an extended J-coloring by the crib graph of a J-colorable graph G. Theorem 3.2. The crib graph c(G) of a J-colourable graph G is also J-colourable. Also, J(c(G)) = J(G) + 1. Proof. Assume that the graph G under consideration admits a J-coloring, say C = fc1; c2; : : : ; ckg, where k = χ(G), the chromatic number of G. While coloring the vertices of c(G), we notice the following points: (i) Since, the twin vertices ui and vi in c(G) are adjacent to each other, both of them can have the same color.
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